Intuitive explanation for the general determinant formula?

Could anyone give an intuitive explanation of the determinant? I know mostly what the determinant means and I can calculate it etc. But I have never got any real-world intuitive explanation of the general formula of the determinant?

How is the formula derived? Where does it come from? What I'm essentially asking is: Prove the general formula for calculating the determinant of an n-by-n matrix and explain the meaning of it

Often the determinant is just defined by the formula:
[tex]\det(a_{ij}) = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{1\sigma(1)} \cdots a_{n\sigma(n)}[/tex]
On the other hand, if you define the determinant as the unique alternating multilinear functional [itex]\det:\mathbb{R}^n \times \cdots \times \mathbb{R}^n \rightarrow \mathbb{R}[/itex] (where the product is taken n-times) satisfying [itex]\det(I) = 1[/itex], then you can recover the formula above for the determinant.

Edit: I suppose this is not really an intuitive explanation, but hopefully it helps a little.

A nice and intuitive explanation of the determinant is that it just represents a signed volume.

For example, given the vectors (a,b) and (c,d) in [itex]\mathbb{R}^2[/itex]. Then we can look at the parallelogram formed by (0,0), (a,b), (c,d) and (a+c,b+d). The area of this parallellogram is given by the absolute value of

Of course the determinant has a sign as well. This is why we call the determinant the signed volume. That is: if we exchange (a,b) and (c,d), then we get the opposite area. The sign is useful for determining orientation.

i suggest you start in 2D and 3D by considering how the determinant relates the the area of a rectangle to the area of the transformed parallelogram or the volume of a cube to the volume of the transformed parallelepiped,

and then how you'd apply that in n dimensions, and how it affects integration after a transformation

saying it is an oriented volume measure implies it should be a multilinear and alternating function. these properties force the formula to be what it is, if you assume the value is 1 on a unit cube. determinants are developed in complete detail starting on p. 62 of these notes.